Question: $\triangle ABC$ and $\triangle DBC$ share $BC$. $AB = 5\ \text{cm}$, $AC = 12\ \text{cm}$, $DC = 8\ \text{cm}$, and $BD = 20\ \text{cm}$. What is the least possible integral number of centimeters in $BC$?

[asy]
size(100); import graph; currentpen = fontsize(10pt);
pair B = (0,0), C = (13,0), A = (-5,7), D = (16,10);

draw(B--A--C--cycle); draw(B--D--C);
label("$A$",A,W); label("$B$",B,W); label("$C$",C,E); label("$D$",D,E);
[/asy]
Solution: By the triangle inequality on triangle $ABC$, $BC > AC - AB = 12 - 5 = 7$, and by the triangle inequality on triangle $BCD$, $BC > BD - CD = 20 - 8 = 12$.  Hence, $BC$ must be at least $\boxed{13}$ centimeters.  (And it is easy to verify that it is possible for $BC$ to be 13 centimeters.